Poisson Hail on a Wireless Ground

Fran\c{c}ois Baccelli; Ke Feng; Sergey Foss

arxiv: 2501.10712 · v2 · submitted 2025-01-18 · 💻 cs.IT · cs.NI· math.IT

Poisson Hail on a Wireless Ground

Fran\c{c}ois Baccelli , Ke Feng , Sergey Foss This is my paper

Pith reviewed 2026-05-23 05:31 UTC · model grok-4.3

classification 💻 cs.IT cs.NImath.IT

keywords wireless networksstability analysiscarrier sensingcollision avoidancePoisson processesMarkov processesqueueing theoryinterference

0 comments

The pith

Carrier sensing and collision avoidance stabilizes wireless networks that would be unstable under immediate greedy access.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a model for wireless systems that combines spatial interference among users, queueing dynamics with users arriving and departing, and WiFi-style carrier sensing where new users wait until nearby transmitters finish. This setup is represented as a Markov process on counting measures under natural assumptions. The main results characterize the stability region via a critical arrival rate and show that sensing can render stable a system that collapses when users access the channel immediately upon arrival. In other words, for typical parameters, the etiquette of waiting makes resource sharing sustainable where greedy behavior does not.

Core claim

For natural values of the system parameters, the implementation of sensing and collision avoidance stabilizes a system that would be unstable if immediate access to the shared resources would be granted. In other words, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

What carries the argument

The Poisson Hail model on a wireless ground, a new Markov process on counting measures that adds carrier sensing and collision avoidance to spatial interference and queueing dynamics.

If this is right

The stability region is bounded by a specific critical arrival rate whose value depends on the sensing rules.
Sensing changes the boundary between stable and unstable regimes compared with immediate-access models.
For parameters typical in wireless settings, the sensing mechanism enlarges the stable region relative to greedy access.
The model unifies spatial birth-and-death processes, Poisson-Hail dynamics, and wireless queueing into a single Markovian framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Protocol designers could use the critical-rate characterization to set sensing thresholds that guarantee stability in dense deployments.
The qualitative result suggests that etiquette rules can substitute for centralized control in interference-limited networks.
Extensions might examine how the stability gain scales with user density or with different interference geometries.
The Markov representation opens the door to computing other performance metrics such as delay or energy use under the same sensing rules.

Load-bearing premise

Under natural assumptions, the system with sensing can be represented as a Markov process on the space of counting measures.

What would settle it

Numerical simulation or analysis of the critical arrival rate in the model with sensing versus the model without sensing, checking whether stability holds exactly where the paper predicts for chosen natural parameter values.

Figures

Figures reproduced from arXiv: 2501.10712 by Fran\c{c}ois Baccelli, Ke Feng, Sergey Foss.

**Figure 2.** Figure 2: An illustration of an underlying deterministic sequence [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: X = [−2, 2]2 . Eh = 1, i.i.d. exponential radius, and function l(r) = min(1, r−4 ), w = 0.05. where c is the constant denoting the maximum Shannon rate achieved when there is no interference as defined earlier. In [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

**Figure 4.** Figure 4: X = [−10, 10]2 . Eh = 1, fixed constant radius, and function l(r) = min(1, r−4 ), w = 0.05. global FIFO, which is too conservative. 6 Spatial Interaction Queuing Processes This section defines a new and general class of spatial queuing processes. Arrivals in these systems are in terms of e.g. a Poisson rain or a renewal rain as above. Arrivals are independently marked with a locus of arrival x in a given c… view at source ↗

read the original abstract

This paper defines a new model which incorporates three key ingredients of a large class of wireless communication systems: (1) spatial interactions through interference, (2) dynamics of the queueing type, with users joining and leaving, and (3) carrier sensing and collision avoidance as used in, e.g., WiFi. In systems using (3), rather than directly accessing the shared resources upon arrival, a customer is considerate and waits to access them until nearby users in service have left. This new model can be seen as a missing piece of a larger puzzle that contains such dynamics as spatial birth-and-death processes, the Poisson-Hail model, and wireless dynamics as key other pieces. It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures. The main results are then two-fold. The first is on the shape of the stability region and, more precisely, on the characterization of the critical value of the arrival rate that separates stability from instability. The second is of a more qualitative or perhaps even ethical nature. There is evidence that for natural values of the system parameters, the implementation of sensing and collision avoidance stabilizes a system that would be unstable if immediate access to the shared resources would be granted. In other words, for these parameters, renouncing greedy access makes sharing sustainable, whereas indulging in greedy access kills the system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a Markov model on counting measures for wireless systems with carrier sensing and claims this stabilizes networks that collapse under greedy access.

read the letter

The punchline is that this paper defines a model combining spatial interference, queueing arrivals and departures, and carrier sensing with collision avoidance, represents the dynamics as a Markov process on counting measures, characterizes the critical arrival rate, and argues that the sensing rule stabilizes regimes unstable under immediate access. What is new is the specific integration of these three pieces, which the abstract positions as previously uncombined in this form. The paper does a clean job stating the two results and drawing the qualitative contrast between considerate and greedy access. The soft spots center on the Markov representation. It rests on natural assumptions that the current configuration of active users suffices to determine future evolution, including waiting times for nearby departures. If service times are general rather than exponential or if the neighborhood rule leaves residual information outside the state, the property fails and the stability claims do not follow. The abstract supplies no derivation or verification of these assumptions, so the full paper must deliver the details without circularity. The evidence for the stabilization effect also needs direct checking against the proofs. This work is for researchers in stochastic geometry and wireless queueing models. Readers already following spatial birth-death processes or Poisson-Hail dynamics will see the most value in the framework and the protocol implication. It deserves a serious referee because the model is original and the claims are specific enough to be checked. I would send it for peer review with attention to the Markov assumption and the comparison result.

Referee Report

2 major / 2 minor

Summary. The paper introduces a model combining spatial interference, queueing dynamics, and carrier sensing/collision avoidance (as in WiFi). It claims that under natural assumptions the system is representable as a Markov process on the space of counting measures. The two main results are (i) a characterization of the stability region via the critical arrival rate and (ii) evidence that, for natural parameter values, collision avoidance stabilizes regimes that are unstable under immediate/greedy access.

Significance. If the Markov representation and stability results hold, the work supplies a missing link between spatial birth-and-death processes, the Poisson-Hail model, and practical wireless protocols. The qualitative stabilization claim has direct implications for the design of considerate access policies.

major comments (2)

[Abstract / modeling section] Abstract and the modeling section: the claim that the dynamics yield a Markov process on counting measures is load-bearing for both the critical-rate characterization and the greedy-vs-avoidance comparison. The state must encode all information needed to determine future waiting times and interference; if service times are general or the definition of “nearby” leaves residual memory, the Markov property fails. Explicit verification of the assumptions that close this gap is required.
[Stability results] Stability-region result: the characterization of the critical arrival rate and the comparison showing stabilization under avoidance both rest on the Markov representation. Without a self-contained proof or counter-example check that the state is sufficient, the load-bearing claims on stability versus instability cannot be assessed.

minor comments (2)

[Introduction] The phrase “natural assumptions” is used repeatedly without an enumerated list; provide a precise bullet list of the required conditions (service-time distribution, interference model, definition of nearby) in the introduction.
Notation for the space of counting measures and the transition rates should be introduced once and used consistently; a short table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The two major comments correctly identify that the Markov representation is foundational to both main results. We address each point below and will revise the manuscript to make the required verifications more explicit and self-contained.

read point-by-point responses

Referee: [Abstract / modeling section] Abstract and the modeling section: the claim that the dynamics yield a Markov process on counting measures is load-bearing for both the critical-rate characterization and the greedy-vs-avoidance comparison. The state must encode all information needed to determine future waiting times and interference; if service times are general or the definition of “nearby” leaves residual memory, the Markov property fails. Explicit verification of the assumptions that close this gap is required.

Authors: We agree that explicit verification is needed. The model assumes exponentially distributed service times (memoryless property) and a fixed interference relation that depends only on the current set of active users; the state is the counting measure of active transmitters. These assumptions close the gap, but the manuscript presents them concisely. In revision we will add a dedicated paragraph in the modeling section that (i) states the assumptions explicitly, (ii) verifies that the future evolution depends only on the current counting measure, and (iii) confirms that no residual memory arises from the carrier-sensing rule. revision: yes
Referee: [Stability results] Stability-region result: the characterization of the critical arrival rate and the comparison showing stabilization under avoidance both rest on the Markov representation. Without a self-contained proof or counter-example check that the state is sufficient, the load-bearing claims on stability versus instability cannot be assessed.

Authors: The stability theorems are proved for the Markov process constructed under the assumptions above. To make the argument fully self-contained we will expand the modeling section with the verification requested in comment 1 and, in the stability section, add a short remark that recalls why the state is sufficient before invoking the standard Foster-Lyapunov or fluid-limit arguments. This will allow readers to assess the claims without external references. revision: yes

Circularity Check

0 steps flagged

No circularity; Markov representation presented as derived result under stated assumptions

full rationale

The provided abstract and description contain no equations, fitted parameters, or self-citation chains that reduce any claimed result to its inputs by construction. The Markov process representation on counting measures is explicitly described as something 'shown' under natural assumptions rather than posited as an unverified axiom or self-definition. The stability region characterization and the qualitative comparison between collision-avoidance and greedy access are presented as main results without any indication that they are forced by prior fits or renamings. No load-bearing uniqueness theorems or ansatzes imported via self-citation are visible. The derivation chain therefore remains self-contained against the given text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5778 in / 974 out tokens · 45267 ms · 2026-05-23T05:31:12.607444+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

It is shown that, under natural assumptions, this model can be represented as a Markov process on the space of counting measures... λc := 2k / (pb |X| Ebσ)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The service rate of each active hailstone... b log(1 + s / (w + Σ l(||xk−xn||)))

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Poisson hail on a hot ground,

F. Baccelli and S. Foss, “Poisson hail on a hot ground,” Journal of Applied Probability, vol. 48, no. A, p. 343–366, 2011

work page 2011
[2]

Spatial birth and death processes,

C. Preston, “Spatial birth and death processes,” Advances in applied probability, vol. 7, no. 3, pp. 465–466, 1975

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Spatial birth–death wireless net- works,

A. Sankararaman and F. Baccelli, “Spatial birth–death wireless net- works,” IEEE Transactions on Information Theory , vol. 63, no. 6, pp. 3964–3982, 2017

work page 2017
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Tse and P

D. Tse and P. Viswanath, Fundamentals of wireless communication . Cambridge university press, 2005

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Interference queueing net- works on grids,

A. Sankararaman, F. Baccelli, and S. Foss, “Interference queueing net- works on grids,” The Annals of Applied Probability , vol. 29, no. 5, pp. 2929–2987, 2019

work page 2019
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A stochastic geometry analysis of dense ieee 802.11 networks,

H. Q. Nguyen, F. Baccelli, and D. Kofman, “A stochastic geometry analysis of dense ieee 802.11 networks,” inProceedings of the 26th IEEE International Conference on Computer Communications , pp. 1199– 1207, 2007

work page 2007
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work page 2024
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On the saturation rule for the stability of queues,

F. Baccelli and S. Foss, “On the saturation rule for the stability of queues,” Journal of Applied Probability , vol. 32, no. 2, pp. 494–507, 1995. 30

work page 1995
[12]

Foss and N

S. Foss and N. Chernova, Methods for Stochastic Stability . Novosibirsk State University Publisher, 2020

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[1] [1]

Poisson hail on a hot ground,

F. Baccelli and S. Foss, “Poisson hail on a hot ground,” Journal of Applied Probability, vol. 48, no. A, p. 343–366, 2011

work page 2011

[2] [2]

Spatial birth and death processes,

C. Preston, “Spatial birth and death processes,” Advances in applied probability, vol. 7, no. 3, pp. 465–466, 1975

work page 1975

[3] [3]

Spatial birth–death wireless net- works,

A. Sankararaman and F. Baccelli, “Spatial birth–death wireless net- works,” IEEE Transactions on Information Theory , vol. 63, no. 6, pp. 3964–3982, 2017

work page 2017

[4] [4]

Tse and P

D. Tse and P. Viswanath, Fundamentals of wireless communication . Cambridge university press, 2005

work page 2005

[5] [5]

Interference queueing net- works on grids,

A. Sankararaman, F. Baccelli, and S. Foss, “Interference queueing net- works on grids,” The Annals of Applied Probability , vol. 29, no. 5, pp. 2929–2987, 2019

work page 2019

[6] [6]

A stochastic geometry analysis of dense ieee 802.11 networks,

H. Q. Nguyen, F. Baccelli, and D. Kofman, “A stochastic geometry analysis of dense ieee 802.11 networks,” inProceedings of the 26th IEEE International Conference on Computer Communications , pp. 1199– 1207, 2007

work page 2007

[7] [7]

Characterizing stability regions in wireless networks with hard-core spatial constraints,

Z. Chen and Y. Zhong, “Characterizing stability regions in wireless networks with hard-core spatial constraints,” IEEE Wireless Commu- nications Letters, vol. 13, no. 8, pp. 2180–2184, 2024

work page 2024

[8] [8]

A mean field analysis of CSMA/CA throughput,

M. Michalopoulou and P. M¨ ah¨ onen, “A mean field analysis of CSMA/CA throughput,” IEEE Transactions on Mobile Computing , vol. 16, no. 8, pp. 2093–2104, 2017

work page 2093

[9] [9]

Spatial network calculus and performance guarantees in wireless networks,

K. Feng and F. Baccelli, “Spatial network calculus and performance guarantees in wireless networks,” IEEE Transactions on Wireless Com- munications, vol. 23, no. 5, pp. 5033–5047, 2024

work page 2024

[10] [10]

The ultimate rank of tropical matrices,

P. Guillon, Z. Izhakian, J. Mairesse, and G. Merlet, “The ultimate rank of tropical matrices,” Journal of Algebra , vol. 437, pp. 222–248, 2015

work page 2015

[11] [11]

On the saturation rule for the stability of queues,

F. Baccelli and S. Foss, “On the saturation rule for the stability of queues,” Journal of Applied Probability , vol. 32, no. 2, pp. 494–507, 1995. 30

work page 1995

[12] [12]

Foss and N

S. Foss and N. Chernova, Methods for Stochastic Stability . Novosibirsk State University Publisher, 2020

work page 2020

[13] [13]

Denisov, Markov chains and random walks with heavy-tailed incre- ments

D. Denisov, Markov chains and random walks with heavy-tailed incre- ments. PhD Thesis, Heriot-Watt University, 2004

work page 2004

[14] [14]

On transient conditions for Markov chains,

S. Foss and D. Denisov, “On transient conditions for Markov chains,” Siberian Mathematical Journal , vol. 42, no. 2, pp. 425–433, 2001

work page 2001

[15] [15]

Asmussen, Applied Probability and Queues

S. Asmussen, Applied Probability and Queues . Springer, 2nd Edition, 2003. 31

work page 2003